Linear Darcy Law Research Articles

Quantitative assessments on fluid flow through fractures embedded in permeable host rocks: Experiments and simulations

An accurate understanding on hydraulic behaviors of fluid flow through fractured rocks is of central importance for both underground resources exploitation and CO2 geological sequestration. Despite its importance, fluid flow through fractures surrounded by permeable matrix is rarely investigated. Here, this study aims to explore fluid flow through fractures surrounded by permeable host rocks using a combined experimental and numerical approach. In particular, core-flooding experiments were sequentially conducted on both intact and fractured cores subjected to different rock types, under varying confining stress and fracture roughness. According to the experimental results, fracture permeability decreases rapidly with increasing confining stress and fracture roughness, while matrix permeability is less affected by confining pressure, resulting in an increment in contributions of matrix flow to the overall flow with increased confining pressure. In addition, numerical simulations were performed on digital fractured rocks with a broadened range of roughness from 2.58 to 18.2, and under confining pressure ranging from 2 MPa to 53 MPa. The fluid flows within matrix and fracture were assumed to follow linear Darcy's law and nonlinear Forchheimer's law, respectively. The simulated results also show that Km/Kf increases with the increase of confining pressure, reflecting the enhancement in contributions of matrix permeability to the overall transportation. If Km/Kf = 0.1 is defined as a threshold over which matrix permeability should not be ignored, then the critical confining pressure of 18 MPa, 20 MPa, and 46 MPa is determined for mudstone, sandstone and limestone, respectively. In addition, the fluid flow through rough-walled fractures is observed to be susceptible to normal loading and fracture roughness, which directly alters flow paths within the fractures. With an increment in confining pressure and fracture roughness, the contact area between rough fracture surfaces grows, promoting flow tortuosity and generating eddies inside fracture, which consequently increases flow resistance and hence weakens fracture permeability. Ultimately, by fitting the simulated data, an empirical equation linking fracture permeability with confining stress, fracture roughness and hydraulic gradient is provided. This study provides quantitative characterizations on fluid flow through fractures accounting for permeable matrix, and thus can potentially improve the accuracy in predicting hydraulic properties of fractured rocks.

Numerical Treatment of State‐Dependent Permeability in Multiphysics Problems

AbstractConstitutive laws relating fluid potentials and fluxes in a nonlinear manner are common in several porous media applications, including biological and reactive flows, poromechanics, and fracture deformation. Compared to the standard, linear Darcy's law, such enhanced flux relations increase both the degree of nonlinearity, and, in the case of multiphysics simulations, coupling strength between processes. While incorporating the nonlinearities into simulation models is thus paramount for computational efficiency, correct linearization, as is needed for incorporation in Newton's method, is challenging from a practical perspective. The standard approach is therefore to ignore nonlinearities in the permeability during linearization. For finite volume methods, which are popular in porous media applications, complete linearization is feasible only for the simplest flux discretization, namely the two‐point flux approximation. We introduce an approximated linearization scheme for finite volume methods that is exact for the two‐point scheme and can be applied to more advanced and accurate discretizations, exemplified herein by a multi‐point flux stencil. We test the new method for both nonlinear porous media flow and several multiphysics simulations. Our results show that the new linearization consistently outperforms the standard approach. Moreover our scheme achieves asymptotic second order convergence of the Newton iterations, in contrast to the linear convergence obtained with the standard approach.

Semi-empirical models for predicting stable water inflow and influence radius of a tunnel considering non-Darcian effect

How to accurately predict water inflow and relevant groundwater influence of a tunnel are classical scientific issues in the field of hydrogeology, which are also of great significance to the construction and operation of underground engineering. This issue is usually investigated based on the assumption of the linear Darcy's law. However, the tunnel inflow is usually featured by high velocities or high pressures, especially in karst regions, which allows the flow regime to easily develop into non-Darcian flow regime. The impact of the non-Darcian effect on predicting water inflow and influence radius of a tunnel has not yet been systematically examined. Here, we numerically investigated the prediction of stable water inflow and influence radius under non-Darcian flow regime. By numerically solving the continuity equation of seepage flow incorporating the nonlinear Forchheimer law, a series of numerical simulations were conducted for stable water inflow into a tunnel by varying key controlling parameters. The quantitative dependencies of stable water inflow on these parameters were figured out, based on which a semi-empirical stable water inflow model was developed considering the non-Darcian effect. Meanwhile, the magnitude of the influence radius was investigated by statistical analysis of 59 actual engineering cases and estimated using existing empirical formulas. Based on these preliminary explorations, a semi-empirical model for estimating influence radius was concluded from massive simulations. The validities of our proposed semi-empirical models for stable water inflow and influence radius were further numerically verified. Comparative study and application in a real engineering case showed that compared with our proposed models, most existing empirical methods predicted a larger stable water inflow and a smaller influence range of groundwater. Besides, comparative results of proposed models with omitting non-Darcian term indicated that neglecting non-Darcian effect would overestimate both the stable water inflow and influence radius. These findings and established models would be useful for assessing stable water inflow and influence radius of a tunnel under non-Darcian flow regime.

ONE CLASS OF SMOOTH BOUNDED SOLUTIONS TO THE CAUCHY PROBLEM FOR A THREE-DIMENSIONAL FILTRATION MODEL WITH DARCY'S LAW

In the filtration theory there are numerous approaches to solving three-dimensional problems of fluid motion in a porous medium. Basically, the solutions of such problems are obtained by numerical methods. The question of finding an analytical solution of precisely three-dimensional problems of fluid motion is open. The first results on the use of the apparatus of four-dimensional mathematics for solving the three-dimensional model of the Navier-Stokes equations by the analytical method were obtained by the Kazakh mathematician Professor M.M. Abenov. After the author of this article with other researchers proved the theorem on the existence of a solution to the Cauchy problem for a three-dimensional model of filtration theory. This paper is devoted to the study of a three-dimensional model of the filtration theory in one of the spaces of four-dimensional numbers. The purpose of this article is to obtain an analytical solution of the three-dimensional Cauchy problem for the mathematical model of linear filtration model by the method of four-dimensional regular functions.  In this study, a class of infinitely differentiable and bounded functions of the initial conditions of the Cauchy problem, satisfying the Cauchy-Riemann condition, with five degrees of freedom for a specific four-dimensional function is found, and also a class of infinitely differentiable and bounded solutions of this problem is found that satisfy the linear Darcy law.

Asymptotic Behavior of a Bingham Flow in Thin Domains with Rough Boundary

We consider an incompressible Bingham flow in a thin domain with rough boundary, under the action of given external forces and with no-slip boundary condition on the whole boundary of the domain. In mathematical terms, this problem is described by non linear variational inequalities over domains where a small parameter \(\epsilon \) denotes the thickness of the domain and the roughness periodicity of the boundary. By using an adapted linear unfolding operator we perform a detailed analysis of the asymptotic behavior of the Bingham flow when \(\epsilon \) tends to zero. We obtain the homogenized limit problem for the velocity and the pressure, which preserves the nonlinear character of the flow, and study the effects of the microstructure in the corresponding effective equations. Finally, we give the interpretation of the limit problem in terms of a non linear Darcy law.

Flat-radial stationary motion of incompressible oil in a uniform horizontal circular formation according to diffrent filtration laws

Relevance. The article considers the issues of flat-radial motion of incompressible oil in a uniform horizontal circular formation. Taking into account that filtration obeys different laws, the research was carried out according to the linear Darcy's law, the generalized Darcy's law and the modified Kesson model. Methodology. Each of the tasks was solved using mathematical methods. The corresponding algorithms were obtained, taking into account the forms of oil movement in a porous medium. Plane-parallel simple filtration flow of oil moves from a strip-like reservoir to a straight gallery. This fluid flow occurs when the oil field under development has several parallel, straight rows of production producing wells. In oil-bearing areas between parallel adjacent rows, oil filtration is also plane-parallel, which implies the practical importance of solving the problem of plane-parallel oil flow in this scientific article. For each filtration law, calculated hydrodynamic formulas for well operation parameters and oil reservoir development indicators are derived. Results. The obtained models of oil flow rate, filtration rate, distribution law of current pressure, current pressure gradient, duration of oil advance in the drainage zone is expedient to use both in drawing up an optimal reservoir development project and for regulating and adjusting the oil recovery process of operating fields. Three stationary-hydrostatic problems are solved, in which the filtration processes obey only a general nonlinear law. All the basic calculation formulas that characterize the filtration processes are derived. By analyzing these formulas, it is possible to identify the nature of the influence of each well parameter and each reservoir development indicator. It is also possible to apply the obtained results to solve vatious theoretical problems of oil field development and when planning new fields development.

Капиллярная пропитка при нелинейном законе фильтрации

In connection with the increasing share of low-permeability oil and gas reservoirs, of particular scientific and practical interest is the study of the characteristics of the flow of filtration processes in reservoirs characterized by non-linear filtration laws. The article presents the results of an analytical study of the features of capillary impregnation of low-permeability reservoirs characterized by non-linear filtration laws. Mathematical modeling of the impregnation processes was carried out under the assumption that the dependence of the filtration rate on the pressure gradient modulus near the zero point can be represented by a power function. It was established that, in contrast to capillary impregnation processes under the traditional linear Darcy law, the intensity of impregnation of low-permeability layers of a productive formation in a nonlinear case depends on the longitudinal pressure gradient in the formation - the larger this gradient, the higher the intensity of impregnation. It was also shown that for nonlinear filtration laws, the capillary impregnation front in a porous medium with a fluid at rest can propagate at a finite speed, which is impossible with linear filtration laws.

Numerical investigation of permeability models for low viscosity magmas: Application to the 2007 Stromboli effusive eruption

Magma permeability is the most important factor controlling the transition between effusive and explosive styles during magma ascent at active volcanoes. When magma permeability is low, gas bubbles in the melt expand as the pressure decreases; above a critical gas volume fraction threshold, magma fragments, generating an explosive eruption. On the contrary, if magma is sufficiently permeable, gas ascends through the conduit towards the surface faster than the magma ascent speed, producing decoupling of gas and magma and reducing the maximum vesicularity. This decoupled flow inhibits fragmentation and leads to either an effusive eruption or quiescent degassing. Accurate modelling of permeability behaviour is therefore fundamental when simulating magma ascent processes. In this work, we compare different permeability models for low viscosity magmas using a 1D steady-state model. We use, as a test case, the 2007 effusive eruption at Stromboli volcano, Italy. We compare the numerical solutions computed using the linear Darcy's law with those obtained using the non-linear Forchheimer relation. Our numerical results show that, using Darcy's law and appropriate permeability models, it is possible to obtain an effusive eruption in agreement with observations. However, we found that, in the shallow conduit, the limit of applicability of Darcy's law (that is the modified Reynolds number Rem < 10) is exceeded due to high gas flow rates. Furthermore, we show that using Forchheimer's law and some parametric expressions for viscous and inertial permeabilities, results can be compatible with an effusive eruption, once appropriate values are chosen. However, one of the parameters required to obtain an effusive eruption, the friction coefficient between gas and melt, is several orders of magnitude lower than that determined from measurements of solid erupted samples. This result requires further experimental verification. We propose that our novel permeability modelling regime is suitable for basaltic volcanism. We highlight that permeabilities derived from studying solid samples are not representative of the actual permeability of a molten magma, at least in the case of low viscosity basaltic magmas. These findings have fundamental implications for the quantification of permeability, modelling of volcanic processes and volcanic eruption dynamics, and the forecasting of volcanic eruptions.

Non‐Darcian flow experiments of shear‐thinning fluids through rough‐walled rock fractures

AbstractUnderstanding non‐Darcian flow of shear‐thinning fluids through rough‐walled rock fractures is of vital importance in a number of industrial applications such as hydrogeology or petroleum engineering. Different laws are available to express the deviations from linear Darcy law due to inertial pressure losses. In particular, Darcy's law is often extended through addition of quadratic and cubic terms weighted by two inertial coefficients depending on the strength of the inertia regime. The relations between the effective shear viscosity of the fluid and the apparent viscosity in porous media when inertial deviations are negligible were extensively studied in the past. However, only recent numerical works have investigated the superposition of both inertial and shear‐thinning effects, finding that the same inertial coefficients obtained for non‐Darcian Newtonian flow applied in the case of shear‐thinning fluids. The objective of this work is to experimentally validate these results, extending their applicability to the case of rough‐walled rock fractures. To do so, flow experiments with aqueous polymer solutions have been conducted using replicas of natural fractures, and the effects of polymer concentration, which determine the shear rheology of the injected fluid, have been evaluated. Our findings show that the experimental pressure loss‐flow rate data for inertial flow of shear‐thinning fluids can be successfully predicted from the empirical parameters obtained during non‐Darcian Newtonian flow and Darcian shear‐thinning flow in a given porous medium.

The streamline flow around a permeable plate in a plane-parallel channel

An analytical solution of the problem of a symmetric streamline flow of an ideal incompressible fluid around a permeable plate in a plane-parallel channel is constructed. The boundary conditions on the plate correspond to the linear Darcy law and to the condition of the controlling action of the pore structure. The effect of the production of a distributed vorticity when a continuous medium flows through a permeable boundary is taken into account. An exact solution is obtained in a form containing a Schwarz integral. The relation between the resistance of the plate and its relative size and porosity is investigated. The result is used to construct a theory of combined permeability. A relation between the hydraulic loss coefficient and the physical parameters of the combined permeability containing porous and perforated elements is obtained in an explicit form.

Dust Removal Experiment and Numerical Simulation with Porous Ceramic Tube

By the use of filtration experiment system with single ceramic tube, cold experiment and numerical simulation was carried out about the dust removal process. The experiment shows that airflow conforms to the linear Darcy law, the pressure drop increases with the increase of filtration velocity and dust load; with the increase of gas pressure, the permeability of porous ceramic decreases. Simulation shows that velocity and pressure drop values increase along the axial height from bottom to top and reach the maximum at the exit. Due to the differences of gas velocity or particle size, the amounts of sedimentary particles are different at the tube wall, larger velocity and smaller size particles trend to be more equispaced when they are trapped on the surface of tube.

Water seepage flow in concrete

The waterproofing durability of concrete is an attentive focus in underground engineering and hydraulic structures. Water seepage law in non-cracked concrete, concrete under tension, straight smooth surface cracks and real cracks in concrete were mainly studied in this paper. And the waterproofing capacity of concrete was analyzed based on the research results. The experimental results indicated that Non-Darcy model was followed in non-cracked concrete weather it bears tensile stress or not. There existed a balance depth in the process of the water permeation. Tensile stress could change the starting pressure gradient (λ) of concrete. With the increase of tension, the λ decreased. When the concrete was cracked, water infiltrated into the concrete through cracks, the linear Darcy Law was followed in this case. To characterize the water seepage law in the real concrete cracks, two parameters τ (tortuosity of cracks) and m (toughness of cracks) were introduced into the Navier–Stokes Equation. And the result indicated the modified Navier–Stokes Equation could describe the seepage law in real concrete cracks well and the value of m in the real concrete cracks was less than 1.15 according to the experimental results.

The heterogeneous dam problem with leaky boundary condition

We study the heterogeneous dam problem, assuming the ux atthe bottoms of the reservoirs obeying to a nonlinear law called leaky boundarycondition. The velocity and the pressure are related by a nonlinear Darcy'slaw. Under a general monotonicity hypothesis on the permeability matrix, weprove that the free boundary is represented locally by graphs of continuousfunctions. We also prove the uniqueness of minimal and maximal solutions.When the ow is given by a linear Darcy law and the permeability matrix issymmetric, we prove the uniqueness of the reservoirs-connected solution.

Fast porous visco-hyperelastic soft tissue model for surgery simulation: Application to liver surgery

Understanding and modeling liver biomechanics represents a significant challenge due to its complex nature. In this paper, we tackle this issue in the context of real-time surgery simulation where a compromise between biomechanical accuracy and computational efficiency must be found. We describe a realistic liver model including hyperelasticity, porosity and viscosity that is implemented within an implicit time integration scheme. To optimize its computation, we introduce the Multiplicative Jacobian Energy Decomposition (MJED) method for discretizing hyperelastic materials on linear tetrahedral meshes which leads to faster matrix assembly than the standard Finite Element Method. Visco-hyperelasticity is modeled by Prony series while the mechanical effect of liver perfusion is represented with a linear Darcy law. Dynamic mechanical analysis has been performed on 60 porcine liver samples in order to identify some viscoelastic parameters. Finally, we show that liver deformation can be simulated in real-time on a coarse mesh and study the relative effects of the hyperelastic, viscous and porous components on the liver biomechanics.

Wave forces acting on a semi-submerged porous circular cylinder

With potential applications in aquaculture as a sea cage or as floating breakwaters, a porous circular-cylinder semi-submerged in waves is investigated under the assumption of linear wave theory. To simplify the problem, the linear Darcy's law is applied to the porous boundary. The problem is solved by means of the eigenfunction expansion. The fluid domain is divided into two regions and different eigenfunctions are used. There are two horizontal porous plates in the inner region, i.e. the cover and bottom of the cylinder. This leads to a complicated ‘dispersion’ relation to determine the eigenvalues, the solutions of which are complex numbers. To solve the radiation problem, some particular solutions are sought to take account of the normal velocity of the body. The hydrodynamic forces are evaluated by integrating the pressure difference on the two sides of body surface. Computed examples are presented to observe the effects of porosity. The Haskind relations are examined for the porous body. It is found that the damping coefficient consists of two parts. In addition to the component of conventional wave-radiating damping, there exists a second component caused by the porous effects.

Soft computing-based nonlinear fusion algorithms for describing non-Darcy flow in porous media

By increasing the velocity of flow in coarse grain materials, local turbulences are often imposed to the flow. As a result, the flow regime through rockfill structures deviates significantly from linear Darcy law; and nonlinear or non-Darcy flow equations will be applicable. Even though the structures of these nonlinear equations have some physical justifications, empirical studies are still necessary to estimate the parameters of these equations amid a great deal of uncertainty inherent to this estimation process. Consequently, none of the current empirical equations alone seem to be able to model the flow process exactly. In recent years, soft computing, in contrast to classical modeling techniques, has been advocated as a hybrid approach to intelligent paradigms such as neural networks, fuzzy logic, and neuro-fuzzy systems that aim to handle the uncertainties and vagueness in such systems. In this paper, we investigate several soft computing paradigms to combine three of the most commonly validated and utilized empirical solutions in the current literature. In this way, the estimates from the three empirical equations drive a soft computing-based fusion algorithm. The results show that soft computing-based approaches provide a powerful paradigm with a strong ability to model reality. Specifically, this paper concludes that cascade correlation neural networks provide the best fusion algorithm with the highest accuracy among the considered conventional alternatives as well as several other soft computing paradigms.

On the dam problem with two fluids governed by a nonlinear Darcy's law

We consider the problem of two fluids flow through a porous medium governed by a nonlinear law. We prove the existence of a weak solution, establish the local Lipschitz continuity of this solution in the zone above the lower fluid, and prove the continuity of the upper free boundary. In the rectangular case, we prove the existence of a monotone solution with respect to the vertical variable, and the continuity of the lower free boundary. Finally, we prove the uniqueness of a monotone solution with respect to $x$ and $y$, when the dam is rectangular and the flow is governed by the linear Darcy law.

On the Permeability Inside the Granular Media

The permeability and resistance characteristics in porous structures with different grain size and shape have been studied experimentally in the paper. Two materials used in the study are crushed gravels and spherical glass balls with four sizes of 5.2mm,7.4mm,12.2mm,and 31.7mm for gravels and 13.0mm,16.0mm,25.0mm,and 35.0mm for balls, respectively. The permeability determined from linear Darcy Law shows a strong dependence on flow velocity in addition to grain sizes and shapes. This is not consistent to the traditional definition of permeability that is only a function of material and fluid. On the other hand, if the resistance force in porous media is quantified by the sum of linear frictional drag and nonlinear form drag suggested by Ward(1964)the turbulent frictional coefficient shows a similar trend as that of Shield Curve. The authors analyze the experimental raw data via former experienced equation(Sollitt and Cross,1972)and compare with a new equation derived according to real physical concept considering total combined drag forces. This indicates that the intrinsic permeability change significantly as the grain size Reynolds number varies and is not constant in a non-uniform flow field such as in a wave field. Results show that porous-structure characteristics like intrinsic permeability and turbulent coefficient tend to stabilize under high Reynolds number, and description with new equation is better than former equation. The characteristics of resistance force in porous media are therefore deserved further study in order to improve related applications on coastal porous structures.

A finite element algorithm for parameter identification of material models for fluid saturated porous media

AbstractIn this contribution an algorithm for parameter identification of geometrically linear Terzaghi–Biot‐type fluid‐saturated porous media is proposed, in which non‐uniform distributions of the state variables such as stresses, strains and fluid pore pressure are taken into account. To this end a least‐squares functional consisting of experimental data and simulated data is minimized, whereby the latter are obtained with the finite element method. This strategy allows parameter identification based on in situ experiments. In order to improve the efficiency of the minimization process, a gradient‐based optimization algorithm is applied, and therefore the corresponding sensitivity analysis for the coupled two‐phase problem is described in a systematic manner. For illustrative purpose, the performance of the algorithm is demonstrated for a slope stability problem, in which a quadratic Drucker–Prager plasticity model for the solid and a linear Darcy law for the fluid are combined. Copyright © 2001 John Wiley &amp; Sons, Ltd.

Analysis of Hydraulic Fracturing of High Permeability Gas Wells to Reduce Non-Darcy Skin Effects

Abstract Hydraulic fracturing has become increasingly popular in high permeability gas reservoirs in order to reduce apparent skin and thus improve well productivity. The remaining post fracture rate dependent skin effect varies from case to case and it is unclear whether this is a result of non-Darcy flow in the fracture, the reservoir or both. This paper presents a study of the effects of reservoir and fracture turbulence in fractured gas wells. A quantification of the fracture length required to eliminate the effects of reservoir turbulence is obtained by means of a numerical study. A similar study with non-Darcy flow in both the reservoir and the fracture results in a correlation of fracture length required to get zero apparent skin at 75﹪ of AOF as a function of reservoir permeability, pressure, fracture conductivity and β factor. Introduction As shown first by Forschheimer(1) flow through porous media deviates from the linear Darcy's law and can be described by a quadratic equation with a non-Darcy flow coefficient β in the nonlinear term. The coefficient β has been correlated to reservoir permeability by numerous authors; in general β decreases with increasing reservoir permeability. Experience has shown that the non-Darcy term becomes more significant in higher permeability gas reservoirs. Generally for reservoir permeabilities below 1 mD there is little effect from non-Darcy flow. In higher permeability reservoirs non-Darcy flow can significantly restrict well production rates and may also affect the pressure transient response. Hydraulic fracturing has become increasingly popular in high permeability gas reservoirs. Normally the goal for hydraulic fracturing high permeability gas wells is to bypass skin damage and thus increase initial flow rates. These 'skin' fractures are generally small in nature and may or may not include high-grade proppants. Prior to performing such a treatment there is often evidence to support a high permeability reservoir and a high apparent positive skin. Unfortunately there is rarely enough data to distinguish if this skin is a true mechanical skin or rate dependent (non-Darcy) skin. Post fracture analysis may show apparent skins ranging from slightly negative to neutral to positive. If a multi-rate post fracture test was performed it is sometimes possible to separate mechanical skin from rate dependent skin. The basic problem with applying the rate dependent skin analysis to a fractured well is that the rate dependent skin was developed based on the assumption of radial homogeneous flow into the wellbore. By adding a hydraulic fracture the flow regime has been altered and the theory breaks down. It is unclear whether the rate dependent skin is occurring in the linear fracture flow or in the reservoir or in both. The primary purpose of this study is to quantify when reservoir turbulence is insignificant for a fractured well. The second purpose is to provide a simple method for predicting post fracture rates in high permeability reservoirs taking into account fracture turbulence. Literature Review The previous work on non-Darcy flow can be divided into two main categories:what is the exact nature of non-Darcy flow andhow to predict when non-Darcy flow will have a significant effect on production.